Ibrahim is a noble name of a great prophet mentioned in Quran with salutation from Allah on him and his progeny سلام على ابراهيم، , but Ibby is a nick name which is completely different and has nothing from the blessings of the name of Ibrahim. This habit of nick names is in many cultures, but we should not give up the blessings of blessed names and use nick names which has no blessed meaning.
We don't say that it is Haraam to use a nick name, but it will deprive you from the blessing of the real name of the prophet. Same as those who leave using the name Mohammad and use Mo, and those who leave the name Mustafa and use Muss or Moss , etc. This practice should not be encouraged among Muslims.
It should be also mentioned that same nick names are used non Muslims, which means that your Muslim identity which comes from name, and other reasons, is given up if you leave using your real blessed name.